Finite Group Databasebeta

← #30669318

Subgroups — #30669318

Order 16768.

1232 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 189 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×D524C211yesmaximal, characteristic, complemented
2C2×#16862221C212yesmaximal, complemented
3C2×#16862216C212yesmaximal, complemented
4C2×C4×C22⋊D131C211yesmaximal, characteristic, complemented
5#22553230C212yesmaximal, complemented
6#22553222C212yesmaximal, complemented
7#22553213C212yesmaximal, complemented
8#22553167C212yesmaximal, complemented
9#380×C131C211yesmaximal, characteristic, complemented
10C24×D131C2212yes—
11C22×D524—28no—
12C22×D524—24no—
13C22×D524C2212yes—
14C22×D524C2212yes—
15C22×D524C2214yescomplemented
16C22×C22⋊D131C2211yescharacteristic
17C22×C4×D131C2211yescharacteristic, complemented
18#16862221—28no—
19#16862216—28no—
20C4×C22⋊D131—24no—
21C2×Dic131⋊C4C2211yescharacteristic, complemented
22C2×C22⋊Dic131C2211yescharacteristic
23C2×(C2×D131)⋊C4C2214yescomplemented
24C2×(C2×D131)⋊C4C2214yescomplemented
25C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
26C2×(C2×D131)⋊C4C2214yescomplemented
27C2×C4×Dic131C2211yescharacteristic, complemented
28C23×C4×C131C2211yescharacteristic
29C2×C4⋊C4×C131C2212yescomplemented
30C2×C22⋊C4×C131C2212yes—
31#95×C131C2212yescomplemented
32C23×D131—28no—
33C23×D131—24no—
34C23×D131—24no—
35C23×D131—24no—
36C23×D131—44no—
37C23×D131C2314yes—
38C23×D131C2311yescharacteristic
39C2×D524—232no—
40C2×D524—28no—
41C2×D524—28no—
42C2×D524—28no—
43C2×D524—24no—
44C2×D524—24no—
45C2×D524—24no—
46C2×D524—44no—
47C2×D524D418yescomplemented
48C2×C22⋊D131—22no—
49C2×C22⋊D131—22no—
50C2×C22⋊D131D414yescomplemented
51C22×Dic131C2311yescharacteristic
52C22×Dic131C2311yescharacteristic
53C22×Dic131C2311yescharacteristic
54C2×C4×D131—24no—
55C2×C4×D131—22no—
56Dic131⋊C4—22no—
57C22⋊Dic131—22no—
58(C2×D131)⋊C4—28no—
59(C2×D131)⋊C4—28no—
60(C2×D131)⋊C4—22no—
61(C2×D131)⋊C4—44no—
62C4×Dic131—22no—
63C24×C131C2311yescharacteristic
64C22×C4×C131—24no—
65C22×C4×C131D414yescomplemented
66C22×C4×C131C2314yes—
67C22×C4×C131C2311yescharacteristic
68C22×C4×C131C2311yescharacteristic
69C4⋊C4×C131—24no—
70C22⋊C4×C131—24no—
71C22×D131—216no—
72C22×D131—21no—
73C22×D131—24no—
74C22×D131—22no—
75C22×D131—24no—
76C22×D131—24no—
77C22×D131—21no—
78C22×D131—24no—
79C22×D131—24no—
80C22×D131—22no—
81C22×D131—24no—
82C22×D131—416no—
83C22×D131—416no—
84C22×D131—48no—
85C22×D131—48no—
86C22×D131C2×D418yes—
87C22×D131C2×D412yescomplemented
88D524—216no—
89D524—216no—
90D524—416no—
91C22⋊D131—28no—
92C2×Dic131—21no—
93C2×Dic131—21no—
94C2×Dic131—21no—
95C2×Dic131—22no—
96C2×Dic131—21no—
97C2×Dic131—22no—
98C2×Dic131C2×D412yescomplemented
99C4×D131—44no—
100C23×C131—22no—
101C23×C131—22no—
102C23×C131C2×D412yes—
103C23×C131C2×D412yes—
104C23×C131#4112yes—
105C23×C131C2411yescharacteristic
106C2×C4×C131—22no—
107C2×C4×C131—28no—
108C2×C4×C131—24no—
109C2×C4×C131—24no—
110C2×C4×C131—24no—
111C2×C4×C131C2×D414yes—
112C2×C4×C131C2×D412yes—
113C2×C4×C131C2×D412yes—
114C22×C131—22no—
115C22×C131—22no—
116C22×C131—24no—
117C22×C131—24no—
118C22×C131#12114yescomplemented
119C22×C131C22×D412yes—
120C22×C131C22×D411yescharacteristic
121C22×C131C22×D411yescharacteristic
122C22×C131C2×#4111yescharacteristic
123C22×C131H(5,𝔽2)12yes—
124C2×D131—28no—
125C2×D131—22no—
126C2×D131—24no—
127C2×D131—28no—
128C2×D131—28no—
129C2×D131—432no—
130C2×D131—41no—
131C2×D131—44no—
132C2×D131—41no—
133C2×D131—44no—
134C2×D131—44no—
135C2×D131—42no—
136C2×D131—44no—
137C4×C131—22no—
138C4×C131—44no—
139C4×C131#12014yescomplemented
140Dic131—22no—
141Dic131—41no—
142Dic131—41no—
143C2×C131—24no—
144C2×C131C2×#12011yescharacteristic
145C2×C131C2×#12111yescharacteristic
146C2×C131#53511yescharacteristic
147C2×C131D4212yes—
148C2×C131#54612yes—
149D131—42no—
150D131—48no—
151C131#240811yescharacteristic, complemented, Sylow
152#2408—1311nomaximal, Sylow
153C23×D4—1311no—
154C2×#121—1312no—
155C2×#120—1312no—
156C2×C4×D4—1311no—
157#394—1312no—
158#392—1312no—
159#390—1312no—
160#380D13111yescharacteristic, complemented
161#379—1312no—
162C25—1312no—
163C22×D4—2628no—
164C22×D4—2624no—
165C22×D4—1311no—
166C22×D4—1312no—
167C22×D4—1314no—
168C22×D4—1312no—
169C23×C4—1311no—
170C23×C4C2×D13111yescharacteristic
171#121—2628no—
172#120—2628no—
173C4×D4—2624no—
174C2×C4⋊C4—1311no—
175C2×C4⋊C4C2×D13112yescomplemented
176C2×C22⋊C4—1311no—
177C2×C22⋊C4—1311no—
178C2×C22⋊C4—1314no—
179C2×C22⋊C4—1314no—
180C2×C22⋊C4—1314no—
181C2×C22⋊C4C2×D13112yes—
182C2×C42—1311no—
183#95C2×D13112yescomplemented
184C24—2628no—
185C24—2624no—
186C24—2624no—
187C24—2624no—
188C24—5244no—
189C24—1314no—
190C24—1311no—
191C24C22×D13111yescharacteristic
192C2×D4—26232no—
193C2×D4—2628no—
194C2×D4—2628no—
195C2×D4—2628no—
196C2×D4—2622no—
197C2×D4—2622no—
198C2×D4—2624no—
199C2×D4—2624no—
200C2×D4—2624no—
201C2×D4—5244no—
202C2×D4—1314no—
203C2×D4—1318no—
204C22×C4—24no—
205C22×C4—2624no—
206C22×C4—2622no—
207C22×C4—1311no—
208C22×C4—1311no—
209C22×C4—1311no—
210C22×C4D52414yescomplemented
211C22×C4C22×D13114yes—
212C22×C4C22×D13111yescharacteristic
213C22×C4C22×D13111yescharacteristic
214C4⋊C4—24no—
215C4⋊C4—2622no—
216C22⋊C4—24no—
217C22⋊C4—2628no—
218C22⋊C4—2628no—
219C22⋊C4—2622no—
220C22⋊C4—2622no—
221C22⋊C4—5244no—
222C42—2622no—
223C23—22no—
224C23—22no—
225C23—26216no—
226C23—2624no—
227C23—2622no—
228C23—2624no—
229C23—2621no—
230C23—2624no—
231C23—2621no—
232C23—2624no—
233C23—2624no—
234C23—2622no—
235C23—2624no—
236C23—52416no—
237C23—52416no—
238C23—5248no—
239C23—5248no—
240C23—1318no—
241C23—1312no—
242C23D4×D13112yes—
243C23C2×D52412yes—
244C23#1452351412yes—
245C23C23×D13111yescharacteristic
246D4—26216no—
247D4—2628no—
248D4—26216no—
249D4—52416no—
250C2×C4—22no—
251C2×C4—28no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—2621no—
256C2×C4—2621no—
257C2×C4—2622no—
258C2×C4—2621no—
259C2×C4—2621no—
260C2×C4—2622no—
261C2×C4—5244no—
262C2×C4—1312no—
263C2×C4D4×D13114yes—
264C2×C4C2×D52412yes—
265C2×C4C2×D52412yes—
266C22—22no—
267C22—24no—
268C22—24no—
269C22—22no—
270C22—2624no—
271C22—2628no—
272C22—2628no—
273C22—2628no—
274C22—2622no—
275C22—52432no—
276C22—5244no—
277C22—5241no—
278C22—5244no—
279C22—5244no—
280C22—5242no—
281C22—5241no—
282C22—5244no—
283C22#1686222114yescomplemented
284C22C2×D4×D13111yescharacteristic
285C22C2×D4×D13112yes—
286C22C22×D52411yescharacteristic
287C22C2×#1452351411yescharacteristic
288C22#1686229212yes—
289C4—22no—
290C4—44no—
291C4—2622no—
292C4—5241no—
293C4—5241no—
294C4#1686221614yescomplemented
295C2—24no—
296C2—5248no—
297C2—5242no—
298C2C2×#1686221611yescharacteristic
299C2C2×#1686222111yescharacteristic
300C2#2255388111yescharacteristic
301C2#2255394612yes—
302C2#2255395312yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#380×C131C2
hypercenter#380D131
hypocenterC131#2408
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #380

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C23×D524C2semidirect
2. C2×#16862221C2semidirect
3. C2×#16862216C2semidirect
4. C2×C4×C22⋊D131C2semidirect
5. #22553230C2semidirect
6. #22553222C2semidirect
7. #22553213C2semidirect
8. #22553167C2semidirect
9. #380×C131C2semidirect
15. C22×D524C22semidirect
17. C22×C4×D131C22semidirect
21. C2×Dic131⋊C4C22semidirect
23. C2×(C2×D131)⋊C4C22semidirect
24. C2×(C2×D131)⋊C4C22semidirect
25. C2×(C2×D131)⋊C4C22semidirect
26. C2×(C2×D131)⋊C4C22semidirect
27. C2×C4×Dic131C22semidirect
29. C2×C4⋊C4×C131C22semidirect
31. #95×C131C22semidirect
47. C2×D524D4semidirect
50. C2×C22⋊D131D4semidirect
65. C22×C4×C131D4semidirect
87. C22×D131C2×D4semidirect
98. C2×Dic131C2×D4semidirect
118. C22×C131#121semidirect
139. C4×C131#120semidirect
151. C131#2408semidirect
160. #380D131semidirect
175. C2×C4⋊C4C2×D131semidirect
183. #95C2×D131semidirect
210. C22×C4D524semidirect
283. C22#16862221semidirect
294. C4#16862216semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C23×D5242
C2×#168622212
C2×#168622162
C2×C4×C22⋊D1312
#225532302
#225532222
#225532132
#225531672
#380×C1312
#2408131